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Full text of "Mathematical And Physical Papers - Iii"

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Bof. In the case of the long pendulum with the brass sphere, the corrected value of A, deduced from the formula (1*77), was equal to about 0*77 of the first approximate value.
I have not considered it necessary to go through all Bessel's experiments, as it was not to be expected that the formula should account for the whole observed decrement. I have only taken four experiments for each kind of pendulum, namely, I. a, 6, e, and / for the long pendulum with the brass sphere; I. c and d and II. c and d for the short pendulum with the brass sphere; XII. a, 6, c, and d for the long pendulum with the ivory sphere, and XII. a', &', c', and d' for the short pendulum with the ivory sphere. The formula (177) gave the following results.
First case,
Log e. rA = -0000759 ; mean error = '0000020.
Second case,
Loge.rA = '0000504; mean error = '0000075.
Third case,
Log e. A = '000631; mean error = "000046. Fourth case,
Log e. A = -000167 ; mean error = '000074.
Now I = rA, and therefore, to get the values of I deduced from experiment, it will be sufficient to divide the numbers above given by the modulus of the common system of logarithms. The theoretical value of I will be got from (169), if we add to k' the correction A&' depending upon the wire. The following are the results:
long p.      short p.      long p.     short p. brass s.     brass s.     ivory s.     ivory s. 1000000 I for sphere alone in
an unlimited mass of fluid,
by theory .....................      67         50         298        222
additional for wire   ............      27           9         114         39
94         59         412        261
1000000 I by experiment...      175        116        1453        384
It appears then that the calculated rate of decrease of the arc amounts on the average to about half the rate deduced from observation. This is about what we might have expected, considering the various circumstances, all tending materially to